**2. Transformational versus other space-group symbols**

There are several systems of space groups naming. They can be divided into three categories in the context of group generations: (i) the *symbolic* one, containing symbols of 'space-group types', like the sequential or the Schőnflies symbols, (ii) the *geometric* one, with the Hermann-Mauguin and the Shubnikov symbols which contain the glide or screw components of symmetry operations given in such a way that the nature of the symmetry elements, their orientation and relative location can be deduced from the symbol, (iii) the *algebraic* category with the Zacharasen, Schmuelli and Hall symbols of complete generators, most convenient for the derivation of 'specific groups'. The Hermann-Mauguin symbols are mainly used in connection with the International Tables for Crystallography for the designation of the conventional space group descriptions. The computer programs transforming these most informative symbols into specific space groups have limited possibilities, are rather complicated, sensitive to changes in space-group symbols and must contain additional conventions, since H-M symbols do not depend on the origin selection. On


– centred cells are described in the *hexagonal axes*. For the space groups with two origin choices, the description with a centre as the origin is chosen.

Having settled the relation between a space-group type and its well established description, other descriptions of the same space-group type are easily identified by a *transformational space-group symbol* (TSG) obtained from the type symbol and explicitly given coordinate system transformation

$$\text{TSG} = \text{type symbol} \left( \mathbf{P}\_{11'} \mathbf{P}\_{21'} \mathbf{P}\_{31'} \mathbf{P}\_{12'} \mathbf{P}\_{22'} \mathbf{P}\_{32'} \mathbf{P}\_{13'} \mathbf{P}\_{23'} \mathbf{P}\_{33} \right) \left( \mathbf{p}\_{1'} \mathbf{p}\_{2'} \mathbf{p}\_{3} \right) \tag{1}$$

where parameters in the first parenthesis describe the unit cell transformation

$$\mathbf{(a',b',c')} = \mathbf{(a,b,c)} \begin{pmatrix} \mathbf{P\_{11}} \ \mathbf{P\_{12}} \ \mathbf{P\_{13}} \\ \mathbf{P\_{21}} \ \mathbf{P\_{22}} \ \mathbf{P\_{23}} \\ \mathbf{P\_{31}} \ \mathbf{P\_{32}} \ \mathbf{P\_{33}} \end{pmatrix} = \begin{pmatrix} \mathbf{a},\mathbf{b},\mathbf{c} \end{pmatrix} \mathbf{P},\tag{2}$$

and the remaining parameters define the origin shift

248 Recent Advances in Crystallography

descriptions

the other side, the algebraic SG symbols lose a clear geometric interpretation and should be selected from the 'economic' point of view – minimal stored data, a simple and non-redundant algorithm generation. The presented transformational approach to naming and deriving

SG No +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 0 P1 P1� P2 P2� C2 Pm Pc Cm Cc P2/m 10 P2�/m C2/m P2/c P2�/c C2/c P222 P222� P2�2�2 P2�2�2� C222� 20 C222 F222 I222 I2�2�2� Pmm2 Pmc2� Pcc2 Pma2 Pca2� Pnc2 30 Pmn2� Pba2 Pna2� Pnn2 Cmm2 Cmc2� Ccc2 Amm2 Aem2 Ama2 40 Aea2 Fmm2 Fdd2 Imm2 Iba2 Ima2 Pmmm Pnnn Pccm Pban 50 Pmma Pnna Pmna Pcca Pbam Pccn Pbcm Pnnm Pmmn Pbcn 60 Pbca Pnma Cmcm Cmca Cmmm Cccm Cmma Ccca Fmmm Fddd 70 Immm Ibam Ibca Imma P4 P4� P4� P4� I4 I4� 80 P4� I4� P4/m P4�/m P4/n P4�/n I4/m I4�/a P422 P42�2 90 P4�22 P4�2�2 P4�22 P4�2�2 P4�22 P4�2�2 I422 I4�22 P4mm P4bm 100 P4�cm P4�nm P4cc P4nc P4�mc P4�bc I4mm I4cm I4�md I4�cd 110 P4�2m P4�2c P4�2�m P4�2�c P4�m2 P4�c2 P4�b2 P4�n2 I4�m2 I4�c2 120 I4�2m I4�2d P4/mmm P4/mcc P4/nbm P4/nnc P4/mbm P4/mnc P4/nmm P4/ncc 130 P4�/mmc P4�/mcm P4�/nbc P4�/nnm P4�/mbc P4�/mnm P4�/nmc P4�/ncm I4/mmm I4/mcm 140 I4�/amd I4�/acd P3 P3� P3� R3 P3� R3� P312 P321 150 P3�12 P3�21 P3�12 P3�21 R32 P3m1 P31m P3c1 P31c R3m 160 R3c P3�1m P3�1c P3�m1 P3�c1 R3�m R3�c P6 P6� P6� 170 P6� P6� P6� P6� P6/m P6�mc P622 P6�22 P6�22 P6�22 180 P6�22 P6�22 P6mm P6cc P6�cm P6�cm P6�m2 P6�c2 P6�2m P6�2c 190 P6/mmm P6/mcc P6�/mcm P6�/mmc P23 F23 I23 P2�3 I2�3 Pm3� 200 Pn3� Fm3� Fd3� Im3� Pa3� Ia3� P432 P4�32 F432 F4�32 210 I432 P4�32 P4�32 I4�32 P4�3m F4�3m I4�3m P4�3n F4�3c I4�3d 220 Pm3�m Pm3�m Pn3�n Pm3�n Fm3�m Fm3�c Fd3�m Fd3�c Im3�m Ia3�d

**Table 1.** The Hermann-Mauguin symbols which can appear in the transformational space-group

the space-group type and points at the 'starting' set of generators.

The transformational approach was inspired by the fact, that in ITA83 all different descriptions of the same space group (settings, origin choices, cell choices) were generated from the same operations transformed to the new coordinate system. Thus, all multiple descriptions as well as any non-conventional description of a given space-group type may be constructed from one set of generators. In contrary to the generally accepted convention that all multiple descriptions are equivalent, one and only one description (and corresponding information: the conventional cell, origin, H-M symbol, etc.) serves as the reference for other SG descriptions. As a result, the number of H-M symbols is reduced to 230 and each such symbol likewise the sequential number or the Schőnflies symbol denotes

The selection of the reference descriptions is based on the following conventions. In monoclinic system the settings with *unique axis b*, *cell choice* 1 are chosen. Five groups with *R*

specific space groups, based on earlier works [10,15], meets the mentioned demands.

$$\mathbf{t} = \mathbf{p}\_1 \mathbf{a} + \mathbf{p}\_2 \mathbf{b} + \mathbf{p}\_3 \mathbf{c}.\tag{3}$$

In most cases some of the parameters are zero and an abbreviated 'short-hand' notation

$$\text{TSG} = \text{type symbol } \text{(a',b',c')} \text{ (t)} \tag{4}$$

is preferred. The shortening concept is similar to that used in storing the symmetry matrices as the coordinate triplets in ITA83. Moreover, the identity transformation (**a**, **b**, **c**) and the zero origin shift (**0**) are omitted in the symbol.

To be familiar with, some examples of the TSG symbols are given underneath:

Fdd2 (b/2+c/2, a/2+c/2, a/2+b/2) – a primitive description of the space group, Pn3� (-1/4,-1/4,-1/4) – *origin choice* 1, 5 (c,a,b) – *unique axis* c, *cell choice* 1, for the group with sequence number 5 or H-M symbol *C*2.

It must be remembered that the Bravais centring letter in an H-M symbol is valid only for the conventional space-group type and generally does not describe the centring vectors resulting from the axis transformation given in the transformational symbols. On the other hand, the use of a sequence number or a Schőnflies symbol to point the space-group type is not very informative. The relation between H-M symbols in the ITA83 and the generators based on the composition series method is lost, but it can be restored by the following trick. Each space group type has one and only one set of generators identified by unique H-M symbol, Schőnflies symbol or even by a sequential number. Thus, the group symbol identifies not only the group type, but also its selected matrix representation (including the axis labelling, the origin). Other group descriptions, conventional or not, need explicit information on the axis transformation and/or the origin shift enabled. In any case an original or transformed generators gives a complete set of symmetry matrices in an unambiguous and effective way.


$$\begin{array}{l} \tilde{\mathbf{x}} = \mathbf{W\_{11}}\mathbf{x} + \mathbf{W\_{12}}\mathbf{y} + \mathbf{W\_{13}}\mathbf{z} + \mathbf{w\_{1}} \\ \tilde{\mathbf{y}} = \mathbf{W\_{21}}\mathbf{x} + \mathbf{W\_{22}}\mathbf{y} + \mathbf{W\_{23}}\mathbf{z} + \mathbf{w\_{2}} \\ \tilde{\mathbf{z}} = \mathbf{W\_{31}}\mathbf{x} + \mathbf{W\_{32}}\mathbf{y} + \mathbf{W\_{33}}\mathbf{z} + \mathbf{w\_{3}} \end{array} \tag{5}$$

$$\mathbf{\dot{x}} = \mathbf{W}\mathbf{x} + \mathbf{w} \xrightarrow{\text{definitions}} (\mathbf{W}, \mathbf{w})\mathbf{x}.\tag{6}$$

$$
\tilde{\mathbf{x}} = \mathcal{U}\mathbf{J}\mathbf{x}.\tag{7}
$$

crystallographic symmetry operations related to the crystallographic coordinate system all elements of the matrix are integers reduced to the values {-1,0,1}, if such coordinate system is based on the shortest lattice vectors or on the Bravais cells. In the latter case, especially simple forms of **W** consisting of three or four non-zero elements are obtained. Therefore, in printed descriptions of space groups, like in ITA83, the space - consuming forms (**W**,**w**) or **W** are equivalently given as shorthand notation of the equation systems (5) and called the *coordinate triplets* ݔǡ ݕǡ or the Jones faithful notation.

There are also some advantages in notation (R|**t**) introduced by Seitz in [25] and adopted by solid-states physicists [26]. In this notation 'R' means a point operation **W** given symbolically and '**t**' is an explicitly given translation, equivalent to **w**. Thus, the diversity of R-symbols, corresponding to a set of different symmetry matrices occurring in the conventional space group descriptions, is reduced to 64 items (48 orthogonal matrices with three non-zero elements occuring in the cubic system and 16 additional matrices with four non-zero elements coming from the oriented hexagonal system). The Seitz symbols are very concise and with the help of multiplication tables they identify the product of symmetry operations

$$\mathbf{R}\begin{pmatrix} \mathbf{R}\_1 \ \mathbf{t}\_1 \end{pmatrix} \begin{pmatrix} \mathbf{R}\_2 \ \mathbf{t}\_2 \end{pmatrix} = \begin{pmatrix} \mathbf{R}\_1 \mathbf{R}\_2 \ \mathbf{l}\mathbf{R}\_1 \mathbf{t}\_2 + \mathbf{t}\_1 \end{pmatrix} = \begin{pmatrix} \mathbf{R} \ \mathbf{t} \end{pmatrix},\tag{8}$$

but for explicit values of components **t**, the **W** matrices symbolized by R must be known. Two special actions R on **t** should be distinguished. Assuming the *k* order of R, that is the equality **W***<sup>k</sup>* = **I** is held, these actions can be distinguished by equations

$$\left(\mathbf{R}\mid\mathbf{t}\_{\text{para}}\right)^{k} = \left(\mathbf{I}\mid k\mathbf{t}\_{\text{para}}\right) \text{and} \left(\mathbf{R}\mid\mathbf{t}\_{\text{ortho}}\right)^{k} = \left(\mathbf{I}\mid\mathbf{0}\right),\tag{9}$$

where vectors **t**para and **t**ortho are mutually perpendicular.

250 Recent Advances in Crystallography

**3. Description of symmetry operations** 

are presented in four different forms:

ii. by a general-position diagram,

system of equations

represented in the matrix form as

1, leads to a more homogenous form

iii. by the diagram(s) of symmetry elements,

Crystallographic groups are groups in the mathematical sense of the word *group*, i.e. they are the sets of elements which fulfil the group conditions. The elements of a space group are the symmetry operations of a crystal, which can be described in several ways. In ITA83 they

The great importance of geometrical intuition and the geometric point of view on space groups, their symmetry elements, H-M symbols, Wyckoff positions, origin specifications is clear from the above list. The diagrams are rather complicated constructions and cannot be derived *on line* for any space group description, contrary to the geometric symbols for each symmetry matrix. In the computer applications an operation symbol may be considered as

For easer understanding of the further material, in the first part of this section the common facts about 'standard' algebraic and geometric descriptions of symmetry operations will be recalled. Next the new geometric symbolism will be proposed. A *dual symbol* of space-group operation is based on a dual symbol of point operation [23] and a point on *geometric elements* closest to the origin [18]. These modifications improve the informative properties and also

In the three-dimensional space the general linear mappings, called also the *affine mappings*, transform a point coordinates *x*, *y*, *z* into the coordinates ��� �� � �̃ of the image point by the

> �� � ����������������� �������������������� �̃� �����������������

> > ����������

For computer applications a more convenient description uses so called augmented matrix **W** which combines the (3x3) matrix **W** with a (3x1) column matrix **w** and a row matrix (0 0 0 1). The augmenting of the coordinate column *x*, *y*, *z* by a fourth dummy coordinate, fixed at

. **x x W** (7)

Contrary to the affine mappings, which preserve the straight lines, planes and parallelism of such objects, a special case called *isometries* preserves also distances and in consequence the volumes and orthogonalities. In this situation the determinant det(**W**) = ±1. Moreover, for

(5)

�������� ��� ���� (6)

i. by a list of symmetry matrices of *general position* in the form of coordinate triplets,

iv. by a list of geometric interpretations of symmetry matrices from (i).

the 'gravity centre' of geometric characterization of a symmetry matrix.

reduce some conventions necessary for the standard symbols to be unique.

�� � �� � �

Recently, the set of geometric symbols for R used in ITA83 was compared with other sets of symbols of point-group operations [24]. Some symbols are purely symbolic, other contain geometric information and are more or less self-defined.

Relations between different spaces, their geometric invariants and group properties of such invariants were 'discovered' before the development of the crystallographic groups. Starting from the Erlangen program announced by Felix Klein in 1872, it is understood that different symmetries represented by the abstract structure-groups are consequences of different geometries and their invariants. Thus, there are two points of view on symmetry: one is algebraic and the second one is geometric. In the case of groups described in ITA83, the three items from the four-element list of equivalent symmetry descriptions possess the geometric nature. The geometric information has played important roles in many aspects of crystallography. The algorithms for the characterization of symmetry operations are the foundation for the Hermann-Maugin symbols. The 'secondary symmetry information' is important in understanding the Wyckoff positions, distinguishing partners in an enantiomorfic pair, finding equivalent descriptions of crystal structures or finding transformations between different algebraic descriptions of the same space group.

The description of a general procedure for deriving symbols for the symmetry operations was included in Chapter 11 of ITA83 [17]. A similar approach was used in [16], different modifications can be found in [7,8]. In all this approaches the basic concepts consist in extracting from **w** (**t** in Seitz notation) its characteristic part **w**g interpreted as screw/glide vectors, finding a set of fixed points of pure rotations or reflections and thus orient and locate so called *geometric element* and find the sense of rotation angle symbolized by a number *n* = 2,3,4 or 6 (rotation angle = 360º/*n*). The scheme of calculation and some critical remarks is given below.

#### **3.1. Type of symmetry operation**

The type of symmetry operation is obtained by a modification of point operation symbols according to the characteristic part of translation vector **w**. It is obvious that the type of point operations is completely determined by a matrix part **W** of (**W**,**w**). Table 2 contains the classification of matrices **W** based on the analysis of their traces and determinants.


**Table 2.** Classification of the point symmetry matrices

A vector **w,** if different from **0**, should be decomposed into orthogonal components: a *glide/screw part* **w**g and a *location part* **w**l. While the first component changes operation types, namely: rotations into screw rotations and reflections into glide reflections, the second one is responsible for shifting the corresponding symmetry element from the origin. The part **w**<sup>g</sup> may be derived by projecting **w** onto the space invariant under **W**, but this needs the metrical information about the coordinate system. The metric-free derivation uses the property (9)

$$\left(\left(\mathbf{W},\mathbf{w}\right)^{k} = \left(\mathbf{W},\mathbf{w}\_{\mathbf{g}} + \mathbf{w}\_{1}\right)^{k} = \left(\mathbf{I},k\mathbf{w}\_{\mathbf{g}}\right) \tag{10}$$

and the location part is the difference **w**l = **w** - **w**g.

#### **3.2. Sense of direction and sense of rotation**

To complete the characterization of **W**, excluding the cases where W represents a two-fold rotation or a reflection, the sense of rotation must be determined. For this purpose and also for obtaining the compatibility between analytical descriptions of axes, a convention which fixes the positive direction must be adopted. Such convention was not explicitly described in [17] and in consequence other conventions are sometimes applied. For example, from the two equivalent descriptions of the same crystallographic direction [11�1�] and [1�11] the latter is positively directed according to the rule applied in [13] and the opposite selection is compatible with geometric descriptions in ITA83. A systematic analysis of 'standard' symbols allowed to specify in [18] six conventions C1 – C6, which together with the algebraic procedures lead to unique symbols. Convention C2, in the detailed form is given in Table 3.


**Table 3.** Selection of the positive direction from a pair [*uvw*] and [���̅�� �. The symbols '-', '0', '+' are used for the positive, zero, negative values, respectively.

According to the convention in Table 3, positive directions characterize the positive product of the non-zero components. Such situation cannot occur if there are only two non-zero elements, which differ in the sign. In this case the rule can be stated as 'a negative component may precede a zero component, but never may follow it'.

Assuming a positive direction of the axis corresponding to a rotation or rotoinversion operation has been determined, one of two commonly used procedures [17, 27] should be applied to obtain the sense of rotation. The Boisen & Gibbs procedure given in [27] is more practical. Having the positive axis direction [*uvw*] derived from **W,** the positive sense of rotation is obtained if one of the following conditions is fulfilled:

$$\text{if } \upsilon = w = 0 \text{ and } \mu \text{W}\_{32} > 0 \text{ or } \text{ } w\text{W}\_{21} - \upsilon \text{ W}\_{31} > 0. \tag{11}$$

#### **3.3. Orientation – location part**

252 Recent Advances in Crystallography

remarks is given below.

property (9)

**3.1. Type of symmetry operation** 

**Table 2.** Classification of the point symmetry matrices

and the location part is the difference **w**l = **w** - **w**g.

**3.2. Sense of direction and sense of rotation** 

The description of a general procedure for deriving symbols for the symmetry operations was included in Chapter 11 of ITA83 [17]. A similar approach was used in [16], different modifications can be found in [7,8]. In all this approaches the basic concepts consist in extracting from **w** (**t** in Seitz notation) its characteristic part **w**g interpreted as screw/glide vectors, finding a set of fixed points of pure rotations or reflections and thus orient and locate so called *geometric element* and find the sense of rotation angle symbolized by a number *n* = 2,3,4 or 6 (rotation angle = 360º/*n*). The scheme of calculation and some critical

The type of symmetry operation is obtained by a modification of point operation symbols according to the characteristic part of translation vector **w**. It is obvious that the type of point operations is completely determined by a matrix part **W** of (**W**,**w**). Table 2 contains the

> tr(**W**) 3 2 1 0 -1 -3 -2 -1 0 1 type 1 6 4 3 2 1� 6� 4� 3� m order *k* 1 6 4 3 2 2 6 4 6 2

A vector **w,** if different from **0**, should be decomposed into orthogonal components: a *glide/screw part* **w**g and a *location part* **w**l. While the first component changes operation types, namely: rotations into screw rotations and reflections into glide reflections, the second one is responsible for shifting the corresponding symmetry element from the origin. The part **w**<sup>g</sup> may be derived by projecting **w** onto the space invariant under **W**, but this needs the metrical information about the coordinate system. The metric-free derivation uses the

,, , *<sup>k</sup> <sup>k</sup>*

To complete the characterization of **W**, excluding the cases where W represents a two-fold rotation or a reflection, the sense of rotation must be determined. For this purpose and also for obtaining the compatibility between analytical descriptions of axes, a convention which fixes the positive direction must be adopted. Such convention was not explicitly described in [17] and in consequence other conventions are sometimes applied. For example, from the two equivalent descriptions of the same crystallographic direction [11�1�] and [1�11] the latter is positively directed according to the rule applied in [13] and the opposite selection is

*k* **Ww Ww w I w gl g** (10)

det(**W**)=1 det(**W**)=-1

classification of matrices **W** based on the analysis of their traces and determinants.

In the classical approaches to the considered topic, the orientation of a geometric element is determined together with its location in one step. Solving of three simultaneous equations of reduced operation (**W**,**w**l), given in a matrix form

$$(\mathbf{W} - \mathbf{I})\mathbf{x}\_{\mathbf{F}} + \mathbf{w}\_{\mathbf{I}} = \mathbf{0},\tag{12}$$

leads to the linear or planar sets of solutions (fixed points) for rotations or reflections, respectively. For a rotoinversion other than a pure inversion, the axis results from the equation

$$\mathbf{x}\left(\mathbf{W}^2 - \mathbf{I}\right)\mathbf{x}\_\mathbf{F} + \mathbf{W}\mathbf{w} + \mathbf{w} = 0.\tag{13}$$

In both cases the set of equations is indeterminate, since det(**W** - **I**) in (12) or det(**W2** - **I**) in (13) are equal zero. This leads to parametric solutions, which forms depend on the mode of calculation.

Although the procedure may also be applied to cases where space groups are given in nonconventional descriptions, there are situations where it is difficult to obtain a unique result even for coordinate triplets taken from ITA83 [18].

For example the matrix equation

$$
\begin{pmatrix} 0 & 0 & \overline{1} \\ \overline{1} & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} \chi \\ \overline{\chi} \\ \overline{z} \end{pmatrix} + \begin{pmatrix} 1/6 \\ 1/3 \\ -1/6 \end{pmatrix} = \begin{pmatrix} \chi \\ \overline{\chi} \\ \overline{z} \end{pmatrix}
$$

gives three solutions presented here as the orientation-location parts:

1. *x*, -*x* + 1/3, -*x* + 1/6 if *x* is treated as the parameter, 2. –*y* + 1/3, *y*, *y* – 1/6 if *y* is selected as the parameter, 3. –*z* + 1/6, *z* + 1/6, *z* if *z* represents the variable parameter.

Of course, these solutions describe the same symmetry axis, but differ in a selected parameter, in a positive direction and in a location point. Unique solutions require special conventions in programmable algorithms, based for example on the *row echelon forms* [9, 27], or in post-calculation standardization.

### **3.4. Pseudo-inverse and the point closest to origin**

Another possibility originates from the linear algebra and from the concept of *pseudo-inverse* matrices. This mathematical formulation allows obtaining single points **x**F from equations (12) and (13). Such unique points have simple interpretation; they represent points from the linear or planar sets closest to origin. The derivation of the pseudo-inverse matrix (**W**-**I**)+ is rather cumbersome, but this idea inspires the new point of view on the location derivation and specification in the geometric descriptions.
